Average Error: 30.2 → 0.7
Time: 21.5s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0126197324348960200623981364742576261051:\\ \;\;\;\;\frac{1}{1 + \cos x} \cdot \left(\left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \frac{1}{\sin x}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{24}, x \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \cos x} \cdot \left(\left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \frac{1}{\sin x}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0126197324348960200623981364742576261051:\\
\;\;\;\;\frac{1}{1 + \cos x} \cdot \left(\left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \frac{1}{\sin x}\right)\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{24}, x \cdot \frac{1}{2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 + \cos x} \cdot \left(\left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \frac{1}{\sin x}\right)\\

\end{array}
double f(double x) {
        double r2934364 = 1.0;
        double r2934365 = x;
        double r2934366 = cos(r2934365);
        double r2934367 = r2934364 - r2934366;
        double r2934368 = sin(r2934365);
        double r2934369 = r2934367 / r2934368;
        return r2934369;
}
↓
double f(double x) {
        double r2934370 = 1.0;
        double r2934371 = x;
        double r2934372 = cos(r2934371);
        double r2934373 = r2934370 - r2934372;
        double r2934374 = sin(r2934371);
        double r2934375 = r2934373 / r2934374;
        double r2934376 = -0.01261973243489602;
        bool r2934377 = r2934375 <= r2934376;
        double r2934378 = 1.0;
        double r2934379 = r2934370 + r2934372;
        double r2934380 = r2934378 / r2934379;
        double r2934381 = r2934379 * r2934373;
        double r2934382 = r2934378 / r2934374;
        double r2934383 = r2934381 * r2934382;
        double r2934384 = r2934380 * r2934383;
        double r2934385 = -0.0;
        bool r2934386 = r2934375 <= r2934385;
        double r2934387 = 0.004166666666666667;
        double r2934388 = 5.0;
        double r2934389 = pow(r2934371, r2934388);
        double r2934390 = r2934371 * r2934371;
        double r2934391 = r2934371 * r2934390;
        double r2934392 = 0.041666666666666664;
        double r2934393 = 0.5;
        double r2934394 = r2934371 * r2934393;
        double r2934395 = fma(r2934391, r2934392, r2934394);
        double r2934396 = fma(r2934387, r2934389, r2934395);
        double r2934397 = r2934386 ? r2934396 : r2934384;
        double r2934398 = r2934377 ? r2934384 : r2934397;
        return r2934398;
}

Error

Bits error versus x

Target

Original30.2
Target0.0
Herbie0.7
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.01261973243489602 or -0.0 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.1

      \[\frac{1 - \cos x}{\sin x}\]
    2. Using strategy rm

    3. Applied clear-num1.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
    4. Using strategy rm

    5. Applied flip--1.5

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}}\]

    6. Applied associate-/r/1.5

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{1 \cdot 1 - \cos x \cdot \cos x} \cdot \left(1 + \cos x\right)}}\]

    7. Applied add-cube-cbrt1.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\sin x}{1 \cdot 1 - \cos x \cdot \cos x} \cdot \left(1 + \cos x\right)}\]

    8. Applied times-frac1.6

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sin x}{1 \cdot 1 - \cos x \cdot \cos x}} \cdot \frac{\sqrt[3]{1}}{1 + \cos x}}\]

    9. Simplified1.2

      \[\leadsto \color{blue}{\left(\frac{1}{\sin x} \cdot \left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right)\right)} \cdot \frac{\sqrt[3]{1}}{1 + \cos x}\]

    10. Simplified1.2

      \[\leadsto \left(\frac{1}{\sin x} \cdot \left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right)\right) \cdot \color{blue}{\frac{1}{1 + \cos x}}\]

    if -0.01261973243489602 < (/ (- 1.0 (cos x)) (sin x)) < -0.0

    1. Initial program 60.0

      \[\frac{1 - \cos x}{\sin x}\]
    2. Using strategy rm

    3. Applied clear-num60.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]

    4. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{24} \cdot {x}^{3} + \frac{1}{240} \cdot {x}^{5}\right)}\]

    5. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{24}, x \cdot \frac{1}{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0126197324348960200623981364742576261051:\\ \;\;\;\;\frac{1}{1 + \cos x} \cdot \left(\left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \frac{1}{\sin x}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{1}{24}, x \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \cos x} \cdot \left(\left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \frac{1}{\sin x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))